n n i=1' n Li=1 2. Assume an unlabelled dataset, {x() ERM}=1, with sample mean, x = 11 x(), and an orthonormal basis set

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N N I 1 N Li 1 2 Assume An Unlabelled Dataset X Erm 1 With Sample Mean X 11 X And An Orthonormal Basis Set 1 (345.85 KiB) Viewed 85 times

n n i=1' n Li=1 2. Assume an unlabelled dataset, {x() ERM}=1, with sample mean, x = 11 x(), and an orthonormal basis set {ulu ERM}-1, where d < m. Rm d j=' d = We have investigated the ‘Projected Variance Maximisation' approach to PCA in which we are interested in finding the d-dimensional subspace spanned by {uble RMuli. ub] = čij Vi, j}}1 for which the sum of the sample variance of the data projected onto this subspace is max- imised. This leads to a formulation of the PCA problem as: d n 1 2 — n j=1 i=1 argmax ££(ulw.xle) – ubil. x)? (1) subject to: ulil. ulöl = dij Vinj We then solved this problem using the eigendecomposition of the sample covariance matrix, C. (a) [2 marks] Show that we can re-write the objective of (1) in terms of the sample covariance matrix, C: C= lgti n where X is the centred data matrix: [(x(1) (x (2) - x)? x)? X = [(x(n) – X)?]